US 20040247054 A1 Abstract A method for the fourth-order, blind identification of at least two sources in a system comprising a number of sources P and a number N of reception sensors receiving the observations, said sources having different tri-spectra. The method comprises at least the following steps: a step for the fourth-order whitening of the observations received on the reception sensors in order to orthonormalize the direction vectors of the sources in the matrices of quadricovariance of the observations used; a step for the joint diagonalizing of several whitened matrices of quadricovariance in order to identify the spatial signatures of the sources. Application to a communication network.
Claims(20) 1. A method of fourth-order, blind identification of two sources in a system including a number of sources P and a number N of reception sensors receiving the observations, the sources having different tri-spectra, comprising the following steps:
a) fourth-order whitening of the observations received on the reception sensors in order to orthonormalize the direction vectors of the sources in the matrices of quadricovariance of the observations used, b) joint diagonalizing of several whitened matrices of quadricovariance to identify the spatial signatures of the sources. 2. The method according to where A
_{Q }is a matrix with a dimension (N^{2}×P) defined by A_{Q}=[(α_{1}{circle over (×)}α_{1}*), . . . , (α_{p}{circle over (×)}α_{p}*)], C_{s}(τ_{1},τ_{2},τ_{3}) is a diagonal matrix with a dimension (P×P) defined by C_{s}(τ_{1},τ_{2},τ_{3})=diag[c_{l}(τ_{1},τ_{2},τ_{3}), . . . , c_{p}(τ_{1},τ_{2},τ_{3})] and c_{p}(τ_{1},τ_{2},τ_{3}) is defined by: c _{p}(τ_{1},τ_{2},τ_{3})=<Cum(s _{p}(t), s _{p}(t−τ _{1})*, s _{p}(t−τ _{2})*, s _{p}(t−τ _{3}))> (5) 3. The method according to Step 1: estimating, through Q;{circumflex over ( )} _{x}, of the matrix Q_{x}, from the L observations x(IT_{e}) using a non-skewed and asymptotically consistent estimator. Step 2: eigen-element decomposition of Q;{circumflex over ( )} _{x}, the estimation of the number of sources P and the limiting of the eigen-element decomposition to the P main components: Q;{circumflex over ( )}_{x}≈E;{circumflex over ( )}_{x}Λ;{circumflex over ( )}_{x}E;{circumflex over ( )}_{x} ^{H}, where Λ;{circumflex over ( )}_{x }is the diagonal matrix containing the P eigenvalues with the highest modulus and E;{circumflex over ( )}_{x }is the matrix containing the associated eigenvectors. Step 3: building of the whitening matrix: T;{circumflex over ( )}=(Λ;{circumflex over ( )} _{x})^{−1/2}E;{circumflex over ( )}_{x} ^{H}. Step 4: selecting K triplets of delays (τ _{1} ^{k},τ_{2} ^{k},τ_{3} ^{k}) where |τ_{1} ^{k}|+|τ_{2} ^{k}|+|τ_{3} ^{k}≠0. Step 5: estimating, through Q;{circumflex over ( )} _{x}(τ_{1} ^{k},τ_{2} ^{k},τ_{3} ^{k}), of the K matrices Q_{x}(τ_{1} ^{k},τ_{2} ^{k},τ_{3} ^{k}). Step 6: computing of the matrices T;{circumflex over ( )} Q;{circumflex over ( )} _{x}(τ_{1} ^{k},τ_{2} ^{k},τ_{3} ^{k}) T;{circumflex over ( )}^{H }and the estimation, by U,{circumflex over ( )}_{sol}, of the unitary matrix U_{sol }by the joint diagonalizing of the K matrices T;{circumflex over ( )} Q;{circumflex over ( )}_{x}(τ_{1} ^{k},τ_{2} ^{k},τ_{3} ^{k}) T;{circumflex over ( )}^{H } Step 7: computing T;{circumflex over ( )} ^{#}U;{circumflex over ( )}_{sol}=[b;{circumflex over ( )}_{1 }. . . b;{circumflex over ( )}_{P}] and the building of the matrices B;{circumflex over ( )}_{1 }sized (N×N). Step 8: estimating, through α;{circumflex over ( )} _{P}, of the signatures aq (1≦q≦P) of the P sources in applying a decomposition into elements on each matrix B;{circumflex over ( )}_{1}. 4. The method according to D(A, Â)=(α_{1}, α_{2}, . . . , α_{P}) (16) where and where d(u,v) is the pseudo-distance between the vectors u and v, such that: 5. The method according to 6. The method according to 7. The method according to 8. The method according to 9. The method according to 10. The use of the method according to 11. The method according to and where d(u,v) is the pseudo-distance between the vectors u and v, such that:
12. The method according to and where d(u,v) is the pseudo-distance between the vectors u and v, such that:
13. The method according to 14. The method according to 15. The method according to 16. The method according to 17. The method according to 18. The method according to 19. The method according to 20. The method according to Description [0001] 1. Field of the Invention [0002] The invention relates especially to a method for the learned or blind identification of a number of sources P that is potentially greater than or equal to the number N of sensors of the reception antenna. [0003] It can be used for example in the context of narrow-band multiple transmission. [0004] It is used for example in a communications network. [0005] It can be applied especially in the field of radio communications, space telecommunications or passive listening to these links in frequencies ranging for example from VLF to EHF. [0006]FIG. 1 is a schematic drawing exemplifying an array of several reception sensors or receivers, each sensor receiving signals from one or more radio communications transmitters from different directions of arrival [0007] Each sensor receives signals from a source with a phase and amplitude that are dependent on the angle of incidence of the source and the position of the sensor. FIG. 2 is a drawing exemplifying the parametrization of the direction of a source. This direction is parametrized by two angles corresponding to the azimuth angle θ and the elevation angle Δ. [0008] 2. Description of the Prior Art [0009] The past 15 years or so have seen the development of many techniques for the blind identification of signatures or source direction vectors, assumed to be statistically independent. These techniques have been developed in assuming a number of sources P smaller than or equal to the number of sensors N. These techniques have been described in the references [1][3][7] cited at the end of the description. However, for many practical applications such as HF radio communications, the number of sources from which signals are received by the sensors is increasing especially with the bandwidth of the receivers, and the number of sources P can therefore be greater than the number of sensors N. The mixtures associated with the sources are then said to be under-determined. [0010] A certain number of methods for the blind identification of under-determined mixtures of narrow-band sources for networks have been developed very recently and are described in the references [2] [7-8] and [10]. The methods proposed in the references [2] and [7-8] make use of information contained in the fourth-order (FO) statistics of the signals received at the sensors while the method proposed in the reference [10] make use of the information contained in one of the characteristic functions of the signals received. However, these methods have severe limitations in terms of the prospects of their operational implementation. Indeed, the method described in the reference [2] is very difficult to implement and does not provide for the identification of the sources having the same kurtosis values (standardized fourth-order cumulant). The methods described in the references [7-8] assume that the sources are non-circular. These methods give unreliable results in practice. Finally, the reference method [10] has been developed solely for mixtures of sources with real (non-complex) values. [0011] The object of the present invention relates especially to a new method for the blind identification of an under-determined mixture of narrow-band sources for communications networks. The method can be used especial y to identify up to N [0012] The invention relates to a method for the fourth-order, blind identification of at least two sources in a system comprising a number of sources P and a number N of reception sensors receiving the observations, said sources having different tri-spectra, wherein the method comprises at least the following steps: [0013] a step for the fourth-order whitening of the observations received on the reception sensors in order to orthonormalize the direction vectors of the sources in the matrices of quadricovariance of the observations used, [0014] a step for the joint diagonalizing of several whitened matrices of quadricovariance in order to identify the spatial signatures of the sources. [0015] The number of sources P is for example greater than the number of sensors N. [0016] The method can be used in a communication network. [0017] The method according to the invention has especially the following advantages: [0018] it enables the identification of a number of sources greater than the number of sensors: [0019] for identical sensors: N [0020] for different sensors (arrays with polarization diversity and/or pattern diversity and/or coupling etc) N [0021] the method is robust with respect to Gaussian noise, even spatially correlated Gaussian noise, [0022] it enables the goniometry of each source identified, using a wavefront model attached to the signature, with a resolution potentially higher than that of existing methods, [0023] it enables the identification of I (N [0024] using a performance criterion, it enables the quantitative evaluation of the quality of estimation of the direction vector of each source and a quantitative comparison of two methods for the identification of a given source, [0025] using a step for the selection of the cyclical frequencies, it enables the processing of a number of sources greater than the number of sources processed by the basic method. [0026] Other features and advantages of the invention shall appear more clearly from the following description along with the appended figures, of which: [0027]FIG. 1 shows an example of a communication network, [0028]FIG. 2 shows parameters of a source, [0029]FIG. 3 is a functional diagram of the method according to the invention, [0030]FIG. 4 shows an example of spatial filtering, [0031]FIGS. 5 and 6 show examples of variations of the performance criterion as a function of the number of samples observed, comparing the performance of the method with two prior art methods. [0032] FIGS. [0033] For a clear understanding of the object of the invention, the following example is given by way of an illustration that in no way restricts the scope of the invention for a radio communications network in a multiple-transmission, narrow-band context, with sources having different tri-spectra (of cumulants). [0034] Each sensor of the network, formed by N receivers, receives a mixture of P narrow-band (NB) sources which are assumed to be statistically independent. On this assumption, the vector of the complex envelopes of the signals at output of the sensors is written as follows:
[0035] where s [0036] It is an object of the invention especially to identify the direction vectors α [0037] From this identification, it is then possible to apply techniques for the extraction of the sources by the spatial filtering of the observations. The blind extraction is aimed especially at restoring the information signals conveyed by the sources in not exploiting any a priori information (during normal operation) on these sources. [0038] Fourth-Order Statistics [0039] The method according to the invention makes use of the fourth-order statistics of the observations corresponding to the time-domain averages Q [0040] where * is the conjugate complex symbol, x [0041] where Q [0042] where A [0043] The expression (4b) has an algebraic structure similar to that of the correlation matrix of the observations used in the algorithm SOBI described in the reference [1]. The notation used here below will be Q Q [0044] It is assumed here below that the number of sources P is such that P≦N [0045] Fourth-Order Whitening Step [0046] The first step of the method according to the invention, called FOBIUM, consists of the orthonormalization, in the matrix of quadricovariance Q Q [0047] where A [0048] The whitening matrix sized (P×N ε [0049] where I [0050] Fourth-Order Identification Step [0051] From the expressions (4b) and (11), it is deduced that: [0052] where W(τ U [0053] where Λ and Π are respectively the unit diagonal matrix and the permutation matrix referred to here above. From the equations (11) and (13), it is possible to deduce the matrix A [0054] where T [0055] The matrix B [0056] In this context, the direction vector α [0057] In brief, the different steps of the method according to the invention include at least the following steps: for L vector observations received in the course of the time: x(lTe) (1≦l ≦L), where T [0058] Estimation [0059] Step 1: The estimation, through Q;{circumflex over ( )} [0060] Stationary and centered case: empirical estimator used in the reference [3]. [0061] Cyclostationary and centered case: estimator implemented in the reference [10]. [0062] Cyclostationary and non-centered case: estimator implemented in the reference [11]. [0063] Whitening [0064] Step 2: The eigen-element decomposition of the estimated matrix of quadricovariance Q;{circumflex over ( )} [0065] Step 3: The building of the whitening matrix: T;{circumflex over ( )}=(Λ;{circumflex over ( )} [0066] Selection of the Triplets [0067] Step 4: The selection of K triplets of delays (τ [0068] Estimation [0069] Step 5: The estimation, through Q;{circumflex over ( )} [0070] Stationary and centered case: empirical estimator used in the reference [3]. [0071] Cyclostationary and centered case: estimator implemented in the reference [10]. [0072] Cyclostationary and non-centered case: estimator implemented in the reference [11]. [0073] Identification [0074] Step 6: The computation of the matrices T;{circumflex over ( )} Q;{circumflex over ( )} [0075] Step 7: The computation of T{circumflex over ( )} [0076] Step 8: The estimation, by α;{circumflex over ( )} [0077] Applications [0078] At the end of the step 8, the method has identified the direction vectors of P non-Gaussian sources having different tri-spectra with same-sign kurtosis values. p<N [0079] Using this information, the method may implement a method of goniometry or a spatial filtering of antennas. [0080] A method of goniometry can be used to determine the direction of arrival of the sources and more precisely the azimuth angle θ [0081]FIG. 4 represents a spatial filtering of antennas for spatial filtering structures. It enables especially the optimizing of reception from one or all the sources present by the spatial filtering of the observations. When several sources are of interest to the receiver, we speak of source separation techniques. When no a priori information on the sources is exploited, we speak of blind techniques. [0082] Verification of the Quality of the Estimates [0083] According to one alternative embodiment, the method comprises a step of qualitative evaluation, for each source, of the quality of identification of the associated direction vector. [0084] This new criterion enables the intrinsic comparison of two methods of identification for the restitution of the signature of a particular source. This criterion, for the identification problem, is an extension of the one proposed in [5] for extraction. It is defined by the P-uplet [0085] where
[0086] and where d(u,v) is the pseudo-distance between the vectors u and v, such that:
[0087] In the simulations of FIGS. 5 and 6, there are P=6 statistically independent sources received on a circular array of N=3 sensors having a radius r such that r/λ=0.55 (λ: wavelength). The six sources are non-filtered QPSK sources having a signal-to-noise ratio of 20 dB with a symbol period T=4T [0088] The incidence values of the sources are such that θ [0089] In the above assumptions, FIG. 5 shows the variation in α [0090]FIG. 6 gives a view, in the same context, of the variations of all the α [0091] Variants of the Cyclical FOBIUM Method [0092]FIGS. 7 and 8 show two examples of variants of the method according to the invention known as the cyclical FOBIUM method. [0093] The idea lies especially in introducing selectivity by the cyclical frequencies into the method presented here above and is aimed especially at the blind identification, with greater processing capacity, of under-determined mixtures of cyclostationary sources. [0094] The major difference between the steps 1 to 8 explained here above and this variant is the implementation of a step for the cyclical isolation of the sources by fourth-order discrimination according to their cyclical frequencies. It is thus possible to separately identify the sources associated with a same fourth-order cyclical parameter without being disturbed by the other sources processed separately. [0095] The two variants shown in FIGS. 7 and 8 can be implemented by reiterating the process of cyclical isolation on the “other sources” with other cyclical parameters. The process of cyclical isolation can be applied several times in a third version illustrated in FIG. 9. [0096] Fourth-Order Cyclical Statistics [0097] The fourth-order cyclical statistics of the observations or sensor signals used are characterized by the matrices of cyclical quadricovariance Q [0098] It can be seen that Q [0099] Where Q [0100] where A [0101] It can be seen that the classic quadricovariance of (6) also verifies that Q Q [0102] In stating that Q {tilde over (Q)} [0103] Whitening Step [0104] The first step of the cyclical method orthonormalizes the columns of the matrices A Q [0105] Where {tilde over (Λ)} [0106] where ({tilde over (Λ)} ε [0107] This expression shows that the matrix {tilde over (T)}B [0108] It is recalled that the whitening matrix T of Q [0109] Step of Cyclical Isolation [0110] From the expressions (28)(29) and (21), it is deduced that:
[0111] When there are P [0112] where the matrices U [0113] Where the matrix {tilde over (C)} T [0114] where the matrices are Π [0115] Similarly, from W [0116] that does not depend on the sources of cyclical parameters (α,τ [0117] where the matrix
[0118] is a diagonal matrix with dimensions (P−P [0119] sized (P−P [0120] In particular, in the first version of this variant, it is possible to carry out the cyclical isolation in α′=0 and ε′=1. Writing W(τ [0121] the equations (34) and (35) become: [0122] [0123] Identification Step [0124] The equations (34) and (36) show that it is possible to identify the unitary matrices Π [0125] and to estimate the right unitary matrix, the joint diagonalizing of the matrices is performed: [0126] To estimate the unitary matrices Π [0127] for 1≦j≦K is performed in jointly diagonalizing the matrices
[0128] and then the matrices
[0129] Knowing Π [0130] From the equations (9) and (29), it is possible to deduce the matrix A [0131] where T [0132] In this context, the direction vector α [0133] Recapitulation of the First Version of the Cyclical Procedure [0134] In short, the steps of the first version of the cyclical method are summarized here below and are applied to L observations x(lTe) (1≦l≦L) of the signals received on the sensors (T [0135] Estimation [0136] Step-1: The estimation of the matrices Q [0137] Stationary and centered case: empirical estimator used in the reference [3]. [0138] Cyclostationary and centered case: estimator implemented in the reference [10]. [0139] Cyclostationary and non-centered case: estimator implemented in the reference [11]. [0140] Whitening [0141] Step 2: The eigen-element decomposition of the estimates of the matrices Q [0142] Step 3: The building of the whitening matrices: T=({tilde over (Λ)} [0143] Step 4: The selection of the cyclical parameters (α,τ [0144] Stationary and centered case: empirical estimator used in the reference [3]. [0145] Cyclostationary and centered case: estimator implemented in the reference [10]. [0146] Cyclostationary and non-centered case: estimator implemented in the reference [11]. [0147] Step-5: The computation of a matrix W [0148] Selection [0149] Step-6: The selection of the K triplets of delays (τ [0150] Estimation [0151] Step-7: The estimation of the K matrices Q [0152] Stationary and centered case: empirical estimator used in the reference [3]. [0153] Cyclostationary and centered case: estimator implemented in the reference [10]. [0154] Cyclostationary and non-centered case: estimator implemented in the reference [11]. [0155] Identification [0156] Step-8: The computation of the matrices T [0157] Step-9: The computation of the matrices T [0158] Step-10: The computation of the unitary matrix U in performing: U=[U [0159] Step-11: The computation of T [0160] Step-12: The estimation of the signatures a, (1≦q≦P) of the P sources in applying a decomposition into elements in each matrix B [0161] Recapitulation of the Second Version of the Cyclical Procedure [0162] The steps of the second version of the cyclical FORBIUM method are summarized here below and are applied to L observations x(lTe) (1≦l≦L) of the signals received on the sensors (T [0163] Estimation [0164] Step-1: The estimation of the matrices Q [0165] Stationary and centered case: empirical estimator used in the reference [3]. [0166] Cyclostationary and centered case: estimator implemented in the reference [10]. [0167] Cyclostationary and non-centered case: estimator implemented in the reference [11]. [0168] Step-2: The eigen-element decomposition of the matrices Q [0169] Step-3: The building of the whitening matrices: T=(Λ [0170] Step-4: The selection of the cyclical parameters (α, τ [0171] Stationary and centered case: empirical estimator used in the reference [3]. [0172] Cyclostationary and centered case: estimator implemented in the reference [10]. [0173] Cyclostationary and non-centered case: estimator implemented in the reference [11]. [0174] Step-5: The computation of a matrix W [0175] Step-6: The selection of K sets of parameters (α [0176] Step-7: The estimation of the K matrices A [0177] Stationary and centered case: empirical estimator used in the reference [3]. [0178] Cyclostationary and centered case: estimator implemented in the reference [10]. [0179] Cyclostationary and non-centered case: estimator implemented in the reference [11]. [0180] Step-8: The computation of the matrices T [0181] Step-9: The computation of the matrices T [0182] Step-10: The computation of the unitary matrix U in performing: U=[U [0183] Step-11: The computation of T [0184] Bibliography [0185] [1] A. Belouchrani, K. Abed—Meraim, J. F. Cardoso, E. Moulines, “A blind source separation technique using second-order statistics”, [0186] [2] J F. Cardoso, “Super-symmetric decomposition of the fourth order cumulant tensor”, [0187] [3] J. F. Cardoso, A. Souloumiac, “Blind beam forming for non-gaussian signals”, [0188] [4] P. Chevalier, G. Benoit, A. Ferréol <<Direction finding after blind identification of sources steering vectors: The blind-maxcor and blind-MUSIC methods>>, EUSIPCO, Trieste, pp 2097-2100, 1996 [0189] [5] P. Chevalier, “Optimal separation of independent narrow-band sources: concept and performance”, [0190] [6] P. Chevalier, A. Ferréol , “On the virtual array concept for the fourth-order direction finding problem”, [0191] [7] P. 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